Coupled fixed point theorems for generalized contractive mappings in partially ordered G-metric spaces

In this paper, we show the existence of a coupled fixed point theorem of nonlinear contraction mappings in complete metric spaces without the mixed monotone property and give some examples of a nonlinear contraction mapping, which is not applied to the existence of coupled fixed point by using the mixed monotone property. We also study the necessary condition for the uniqueness of a coupled fixed point of the given mapping. Further, we apply our results to the existence of a coupled fixed point of the given mapping in partially ordered metric spaces. Moreover, some applications to integral equations are presented. MSC: 47H10, 54H25. In this paper, we establish some coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings having the mixed monotone property in partially ordered G-metric spaces. The results on fixed point theorems are generalizations of the recent results of Choudhury and Maity (Math. Comput. Model. 54: 73-79, 2011) and Luong and Thuan (Math. Comput. Model. 55: 1601-1609, 2012).